A continuum damage mechanics method for fracture simulation of quasi‐brittle materials

This paper addresses a novel continuum damage‐based method for simulating failure process of quasi‐brittle materials starting from local damage initiation to final fracture. In the developed method, the preset characteristic length field is used to evaluate damage instead of element, which is used to reduce the spurious sensitivity. In addition, damage is only updated in the most dangerous location at a time for considering stress redistribution due to damage evolution, which is used to simulate competitive fracture process. As cases study, representative numerical simulations of two benchmark tests are given to verify the performance of the developed continuum damage‐based method together with a used damage model. The simulation results of the crack paths for two concrete specimens obtained from the developed method matched well with the corresponding experimental results. The results show that the developed continuum damage‐based method is effective and can be used to simulate damage and fracture process of brittle or quasi‐brittle materials. And the simulation results based on the developed method depend only the preset characteristic length field and not grid mesh.     

Incremental‐secant modulus iteration scheme and stress recovery for simulating cracking process in quasi‐brittle materials using XFEM

In this paper, an incremental‐secant modulus iteration scheme using the extended/generalized finite element method (XFEM) is proposed for the simulation of cracking process in quasi‐brittle materials described by cohesive crack models whose softening law is composed of linear segments. The leading term of the displacement asymptotic field at the tip of a cohesive crack (which ensures a displacement discontinuity normal to the cohesive crack face) is used as the enrichment function in the XFEM. The opening component of the same field is also used as the initial guess opening profile of a newly extended cohesive segment in the simulation of cohesive crack propagation. A statically admissible stress recovery (SAR) technique is extended to cohesive cracks with special treatment of non‐homogeneous boundary tractions. The application of locally normalized co‐ordinates to eliminate possible ill‐conditioning of SAR, and the influence of different weight functions on SAR are also studied. Several mode I cracking problems in quasi‐brittle materials with linear and bilinear softening laws are analysed to demonstrate the usefulness of the proposed scheme, as well as the characteristics of global responses and local fields obtained numerically by the XFEM. Copyright © 2006 John Wiley & Sons, Ltd.     

Why do cracks branch? A peridynamic investigation of dynamic brittle fracture

In this paper we review the peridynamic model for brittle fracture and use it to investigate crack branching in brittle homogeneous and isotropic materials. The peridynamic simulations offer a possible explanation for the generation of dynamic instabilities in dynamic brittle crack growth and crack branching. We focus on two systems, glass and homalite, often used in crack branching experiments. After a brief review of theoretical and computational models on crack branching, we discuss the peridynamic model for dynamic fracture in linear elastic–brittle materials. Three loading types are used to investigate the role of stress waves interactions on crack propagation and branching. We analyze the influence of sample geometry on branching. Simulation results are compared with experimental ones in terms of crack patterns, propagation speed at branching and branching angles. The peridynamic results indicate that as stress intensity around the crack tip increases, stress waves pile-up against the material directly in front of the crack tip that moves against the advancing crack; this process “deflects” the strain energy away from the symmetry line and into the crack surfaces creating damage away from the crack line. This damage “migration”, seen as roughness on the crack surface in experiments, modifies, in turn, the strain energy landscape around the crack tip and leads to preferential crack growth directions that branch from the original crack line. We argue that nonlocality of damage growth is one key feature in modeling of the crack branching phenomenon in brittle fracture. The results show that, at least to first order, no ingredients beyond linear elasticity and a capable damage model are necessary to explain/predict crack branching in brittle homogeneous and isotropic materials.     

A probabilistic nonlocal model for crack initiation and propagation in heterogeneous brittle materials

A probabilistic damage model is developed to study crack initiation and growth in quasi‐brittle materials. Two different thresholds are considered to describe these mechanisms. A Weibull model is used to account for the randomness of crack initiation(s) and then a fracture mechanics based threshold is considered to model crack propagation. The model is integrated in a finite element code via a nonlocal damage approach. A regularization operator based on a stress regularization is introduced. Both damage thresholds are checked using the ‘regularized’ stress field to avoid mesh dependence. The interaction between propagating cracks and potential initiation sites is accounted for. Copyright © 2012 John Wiley & Sons, Ltd.     

Fracture Behaviour Investigation into a Polymer‐Bonded Explosive

Abstract: Polymer‐bonded explosive (PBX) is used widely in weapon systems. Failure of PBX caused by mechanical damage is one of the sources of accidental ignitions. A brittle crack of PBX produces local heating, creating a ‘hot spot’ finally. Investigation into the tensile fracture behaviour of PBX is one of the main works to determine the failure mechanism. Although many researchers have carried out the quasi‐static Brazilian test to understand the damage evolution of PBX, the fracture feature of PBX under dynamic impact is rarely reported. In this article, dynamic Brazilian tests were conducted. A single‐pulse loading apparatus was used to ensure that specimen was loaded only once during a dynamic Brazilian test. High‐speed camera, digital image correlation and micro‐observation techniques were adopted for strain measurement and microfracture observation. All the dynamic tensile crack exhibits transgranular fracture, which indicates more heat would released by the propagation of crack and more friction between fractured crystal surfaces. On the basis of the theories of interface debonding and transgranular fracture, larger crystals are more prone to crack, whereas smaller crystals simply debond with neighbouring binders. Discrete element method simulation results show that specimen with interface debonding microcracks was able to sustain additional load until transgranular fracture begins.     

Two‐scale diffusion–deformation coupling model for material deterioration involving micro‐crack propagation

In this paper, we introduce a two‐scale diffusion–deformation coupled model that represents the aging material deterioration of two‐phase materials involving micro‐crack propagations. The mathematical homogenization method is applied to relate the micro‐ and macroscopic field variables, and a weak coupling solution method is employed to solve the two‐way coupling phenomena between the diffusion of scalar fields and the deformation of quasi‐brittle solids. The macroscopic mechanical behavior represented by the derived two‐scale two‐way coupled model reveals material nonlinearity due to micro‐scale cracking induced by the scalar‐field‐induced deformation, which can be simulated by the finite cover method. After verifying the fundamental validity of the proposed model and the analysis method, we perform a simple numerical example to demonstrate their ability to predict aging material deterioration. Copyright © 2010 John Wiley & Sons, Ltd.     

Stress field interaction and strain energy distribution between a stationary main crack and its process zone

The damage process zone developed by brittle materials in front of a macrocrack is simulated by means of a distribution of microcracks. Crack mutual interactions are taken into account by means of a numerical technique, based on a displacement discontinuity boundary element method that is able of considering both the macrocrack–microcrack and microcrack–microcrack interactions inside the process zone. In the frame of linear elastic fracture mechanics the stress field at each crack tip and the related elastic strain energy are calculated. The main features of the interaction phenomena turn out to be almost independent of the microcrack density. Some considerations both on the shielding and amplification effects on the main crack and on the strain energy distribution between cracks give explanation to experimental evidence and prove that crack interaction is not such a short-range effect as sometimes expected.     

A continuous–discontinuous approach to simulate physical degradation processes in porous media

A macroscopic framework for the simulation of physical degradation processes in quasi‐brittle porous materials is proposed. The framework employs the partition of unity (PU) concept and introduces a cohesive zone model, capturing the entire failure process starting from the growth and coalescence of micro‐defects until the formation of macro‐cracks. The framework incorporates the interaction between the failure process and the heat and mass transfer in the porous medium. As an example, physical degradation of an outside render is studied. The analysis illustrates that both material and interface failure can be investigated with this formulation. Depending on the boundary conditions, either one dominant crack or a network of small cracks is formed. Copyright © 2010 John Wiley & Sons, Ltd.     

Experimental Investigations of Fracture Process in Concrete by Means of X‐ray Micro‐computed Tomography

The paper describes investigation results on fracture in notched concrete beams under quasi‐static three‐point bending by the X‐ray micro‐computed tomography. The two‐dimensional (2D) and three‐dimensional image procedures were used. Attention was paid to width, length, height and shape of cracks along beam depth. In addition, the displacements on the surface of concrete beams during the deformation process were measured with the 2D digital image correlation technique in order to detect strain localisation before a discrete crack occurred. The 2D fracture patterns in beams were numerically simulated with the finite‐element method using an isotropic damage constitutive model enhanced by a characteristic length of micro‐structure. Concrete was modelled as a random heterogeneous four‐phase material composed of aggregate, cement matrix, interfacial transitional zones and air voids. The advantages of the X‐ray micro‐computed tomography were outlined.     

Extrinsic cohesive modelling of dynamic fracture and microbranching instability in brittle materials

Dynamic crack microbranching processes in brittle materials are investigated by means of a computational fracture mechanics approach using the finite element method with special interface elements and a topological data structure representation. Experiments indicate presence of a limiting crack speed for dynamic crack in brittle materials as well as increasing fracture resistance with crack speed. These phenomena are numerically investigated by means of a cohesive zone model (CZM) to characterize the fracture process. A critical evaluation of intrinsic versus extrinsic CZMs is briefly presented, which highlights the necessity of adopting an extrinsic approach in the current analysis. A novel topology‐based data structure is employed to enable fast and robust manipulation of evolving mesh information when extrinsic cohesive elements are inserted adaptively. Compared to intrinsic CZMs, which include an initial hardening segment in the traction–separation curve, extrinsic CZMs involve additional issues both in implementing the procedure and in interpreting simulation results. These include time discontinuity in stress history, fracture pattern dependence on time step control, and numerical energy balance. These issues are investigated in detail through a ‘quasi‐steady‐state’ crack propagation problem in polymethylmethacrylate. The simulation results compare reasonably well with experimental observations both globally and locally, and demonstrate certain advantageous features of the extrinsic CZM with respect to the intrinsic CZM. Copyright © 2007 John Wiley & Sons, Ltd.     

CDM based modelling of damage and fracture mechanisms in concrete under tension and compression

Anisotropic damage evolution and crack propagation in the elastic–brittle materials is analysed by the concepts of continuum damage mechanics (CDM) and finite element method (FEM). The modified Murakami–Kamiya (MMK) model of elastic-damage material is used to describe damage anisotropy in concrete. The Helmholtz free energy representation is discussed. The unilateral crack opening/closure effect is incorporated in such a way that the continuity requirement during unloading holds. The incremental form of the stress–strain equations is developed. The general failure criterion is proposed by checking the positive definiteness of the Hessian matrix of the free energy function. The local approach to fracture (LAF) by FEM is applied to the pre-critical damage evolution that precedes the crack initiation, and the post-critical damage/fracture interaction. Crack is modelled as the assembly of failed finite elements in the mesh, the stiffness of which is reduced to zero when the critical points at stress–strain curves are reached. A concrete specimen with the pre-load, inclined crack is analysed in order to simulate different fracture mechanisms in tension or compression. The constitutive model is capable of predicting the kinked-type crack under tension and the wing-type crack under compression.     

基于拓展虚内键本构模型的多裂纹混凝土结构破坏分析

拓展虚内键(Augmented virtual internal bond, AVIB)是基于虚内键理论的一种多尺度本构模型,它同时考虑了微观虚内键的法向和切向变形,应用Xu-Needleman势函数描述虚内键,并在微观势函数基础上直接导出了宏观本构方程。由于脆性材料的抗压强度与微元体的应力状态有关,为了反映这种微元应力效应,依据混凝土三轴抗压强度准则定义了应力效应系数,并将其反映到AVIB本构模型中。对于已有裂纹,采用无厚度单元劈裂法进行建模,避免了网格重划分问题和单独设置接触单元问题。结合AVIB模型与无厚度单元劈裂法,对多裂纹混凝土结构的破坏进行了模拟分析。结果表明,预制裂纹的长度不同,导致结构的主裂纹扩展方式不同。模拟所得的结构破坏模式及荷载-主裂纹口张开位移曲线与相关文献报道结果基本一致,表明了该方法的有效性。由于该文所采用的AVIB本构方程中已蕴涵了混凝土断裂能及三轴强度准则,因而在整个断裂模拟过程中,既避免了外部的断裂准则问题,同时又不需要网格重构及附加自由度,提高了计算效率,为大体积混凝土结构的破坏分析提供了一种简单的可行方法。     

Failure analysis of quasi‐brittle materials involving multiple mechanisms on fractured surfaces

We present a variational formulation for the quasi‐static boundary value problem of a structure with quasi‐brittle materials, involving (i) unknown states of contact, (ii) deformation‐dependent frictional forces, (iii) crack opening and closing with cohesive traction, and (iv) configuration change due to the initiation and the evolution of cracks, and propose a new finite cover method (FCM) capable of reflecting those multiple mechanisms in the failure analysis. The cover‐division strategy is also introduced to judge the generation of cracks, and to locate and orient them within the framework of the FCM. A relevant numerical algorithm is designed to be consistent with the mathematical representation of the multiple mechanisms. Several numerical examples are presented to validate the proposed method and to demonstrate its promise and potential for evaluating the ultimate strength of quasi‐brittle materials. Copyright © 2006 John Wiley & Sons, Ltd.     

Combined continuum damage‐embedded discontinuity model for explicit dynamic fracture analyses of quasi‐brittle materials

In this paper, a novel constitutive model combining continuum damage with embedded discontinuity is developed for explicit dynamic analyses of quasi‐brittle failure phenomena. The model is capable of describing the rate‐dependent behavior in dynamics and the three phases in failure of quasi‐brittle materials. The first phase is always linear elastic, followed by the second phase corresponding to fracture‐process zone creation, represented with rate‐dependent continuum damage with isotropic hardening formulated by utilizing consistency approach. The third and final phase, involving nonlinear softening, is formulated by using an embedded displacement discontinuity model with constant displacement jumps both in normal and tangential directions. The proposed model is capable of describing the rate‐dependent ductile to brittle transition typical of cohesive materials (e.g., rocks and ice). The model is implemented in the finite element setting by using the CST elements. The displacement jump vector is solved for implicitly at the local (finite element) level along with a viscoplastic return mapping algorithm, whereas the global equations of motion are solved with explicit time‐stepping scheme. The model performance is illustrated by several numerical simulations, including both material point and structural tests. The final validation example concerns the dynamic Brazilian disc test on rock material under plane stress assumption. Copyright © 2014 John Wiley & Sons, Ltd.     

FATIGUE CRACK GROWTH LAWS FOR BRITTLE AND DUCTILE MATERIALS INCLUDING THE EFFECTS OF STATIC MODES AND ELASTIC-PLASTIC DEFORMATION

A fatigue crack growth damage accumulation model is used to derive laws for the fatigue crack growth rates of brittle and ductile materials. The damage accumulated during cyclic loading is assumed to be proportional to the cyclic change in the plastic displacement in the crack tip yielded zone. The static mode contribution to the fatigue damage is assumed to be proportional to some power of the crack tip displacement. The laws are applicable in either the small or large scale yielding regimes provided that the stress ratio remains positive. Static modes are assumed to be controlled by the fracture toughness value in brittle materials, and by the gradient of the crack growth resistance curve in ductile materials. In the analysis of ductile materials it is assumed that the crack growth resistance of the material is not significantly altered by fatigue crack growth.
The growth rate equations are expressed in terms of the near field value of the J -integral, i.e. the value which would be calculated from assuming the material deformed in a non-linear elastic manner during the increasing load part of the fatigue cycle. Examples are given of the predictions of the growth law for ductile materials. It is predicted that after the initiation of stable tearing the crack growth rate, when expressed in terms of the cyclic change in the stress intensity factor, depends on both the structural geometry and the degree of crack tip plastic deformation. In both brittle and ductile materials the fatigue crack growth rate is predicted to accelerate as the failure criteria relevant to static crack instability are approached.     

The influence of higher order terms of Williams series on a more accurate description of stress fields around the crack tip

In this work, the crack tip stress field is investigated in two different configurations loaded in mode I as well as in mode I + II. The detailed two‐dimensional solution includes a conventional finite elements analysis that is compared to results obtained analytically from what is referred to as Williams expansion. The influence of consideration of various numbers of terms of the series expansion on the stress distribution is discussed, and the significance of the multi‐parameter fracture mechanics approach is emphasized for some engineering applications, for example, failure of quasi‐brittle materials and estimation of plastic zone extent.     

Localizing gradient damage model with decreasing interactions

Nonlocal integral and/or gradient enhancements are widely used to resolve the mesh dependency issue with standard continuum damage models. However, it is reported that whereas the structural response is mesh independent, a spurious damage growth is observed. Accordingly, a class of modified nonlocal enhancements is developed in literature, where the interaction domain increases with damage. In this contribution, we adopt a contrary view that the interaction domain decreases with damage. This is motivated by the fact that the fracture of quasi‐brittle materials typically starts as a diffuse network of microcracks, before localizing into a macroscopic crack. To ensure thermodynamics consistency, the micromorphic theory is adopted in the model development. The ensuing microforce balance resembles closely the Helmholtz expression in a conventional gradient damage model. The superior performance of the localizing gradient damage model is demonstrated through a one‐dimensional problem, as well as mode I and II failure in plane deformation. For all three cases, a localized deformation band at material failure is obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

An adaptive material point method coupled with a phase-field fracture model for brittle materials

In this study, a new adaptive method for crack propagation analysis is developed by using the material point method coupled with a phase-field fracture model for brittle materials. A background grid of material particles is adaptively refined based on the amount of material damage to resolve the length scale in the phase-field evolution equation. A division process of the material particles associated with the refined background cells is also performed to increase the resolution of solutions near the crack tip. The effectiveness and validity of the proposed method is assessed through several numerical examples for crack propagation in brittle materials.